My question is that how would one count the number of ways to distribute things from a group of different things and different groups of alike things (for e.g., $a$ alike things of one kind, $b$ alike things of separate same kind and so on..) into a specific number of groups containing say $n_1$ elements $n_2$ elements and so on. Kindly tell me if this distribution formula which you give me in the end changes if distribution is to be done in same groups i.e. $m$ groups containing same elements.
eg: say we have {a,a, b,b,b,b, c,d,e} and I want to divide them in groups of 3 elements each. Orientation of elements in a group does not matter ,ie order of elements in a group doesn't matter. or say a simpler eg (which I have calculated)
is say you have {a,a,b,b,c,c} and you want to divide them in 3 groups of 2 elements each so the elements will be
{(a,a)(b,b)(c,c)}
{(a,a)(b,c)(b,c)}
{(a,b)(a,b)(c,c)}
{(a,b)(a,c)(b,c)}
{(a,c)(a,c)(b,b)}
these are the only 4 possibilities i repeat that the order of elements in the group does not matter ie (a,b) and (b,a) are one and same thing i am just looking for different arrangement.